§19.4(ii) Differential Equations

Notes:

See Cazenave (1969, p. 175). (19.4.8) agrees also
with Edwards (1954, vol. 1, p. 402) and with expansion to first
order in k. The term on the right side in Byrd and Friedman (1971, 118.01)
has the wrong sign.

If ϕ=π/2, then these two equations become hypergeometric
differential equations (15.10.1) for K⁡(k) and
E⁡(k). An analogous differential equation of third order for
Π⁡(ϕ,α2,k) is given in Byrd and Friedman (1971, 118.03).